It looks like some cards need the special one with the aux. power input... but other than that... for the purpose of mining BTC, there should be no reason why any card wouldn't work over a 1x slot.

The 1x is all about the baud. A 16x slot can transfer data 16 times faster than a 1x slot... but, since PCI-e allows the bus and devices to negotiate their own speeds, even a high speed card should be able to negotiate down to a slower transfer rate.